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MATHEMATICS-I

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CONTENTS

Ordinary Differential Equations of First Order and First Degree

Linear Differential Equations of Second and Higher Order

Mean Value Theorems

Functions of Several Variables

Curvature, Evolutes and Envelopes

Curve Tracing Applications of Integration

Multiple Integrals

Series and Sequences

Vector Differentiation and Vector Operators

Vector Integration Vector Integral Theorems

Laplace transforms

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TEXT BOOKS

A text book of Engineering Mathematics, Vol-IT.K.V.Iyengar, B.Krishna Gandhi and Others,S.Chand & Company

A text book of Engineering Mathematics,C.Sankaraiah, V.G.S.Book Links

A text book of Engineering Mathematics, Shahnaz ABathul, Right Publishers

A text book of Engineering Mathematics,P.Nageshwara Rao, Y.Narasimhulu & N.PrabhakarRao, Deepthi Publications

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REFERENCES

A text book of Engineering Mathematics,

B.V.Raman, Tata Mc Graw Hill

Advanced Engineering Mathematics, Irvin

Kreyszig, Wiley India Pvt. Ltd.

A text Book of Engineering Mathematics,

Thamson Book collection

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UNIT-I

ORDINARY DIFFERENTIAL

EQUATIONS OF FIRST ORDERAND FIRST DEGREE

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UNIT HEADER

Name of the Course:B.Tech

Code No:07A1BS02

Year/Branch:I YearCSE,IT,ECE,EEE,ME,CIVIL,AERO

Unit No: I

No.of slides:26

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S.No. Module LectureNo.

PPT Slide No.

1 Introduction,Exact

differential equations

L1-10 8-19

2 Linear and Bernoullisequations,Orthogonal

trajectories

L11-13 20-23

3 Newtons law ofcooling and decay

L14-15 24-26

UNIT INDEX

UNIT-I

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Lecture-1

INTRODUCTION

An equation involving a dependent variable

and its derivatives with respect to one or more

independent variables is called a Differential

Equation.

Example 1: y + 2y = 0

Example 2: y2

2y1

+y=23

Example 3: d2y/dx2 + dy/dxy=1

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TYPES OF A DIFFERENTIAL EQUATION

ORDINARY DIFFERENTIAL EQUATION: A

differential equation is said to be ordinary, if the

derivatives in the equation are ordinary derivatives.

Example:d2y/dx2-dy/dx+y=1 PARTIAL DIFFERENTIAL EQUATION: A

differential equation is said to be partial if the

derivatives in the equation have reference to two or

more independent variables.

Example:4y/x4+y/x+y=1

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Lecture-2

DEFINITIONS

ORDER OF A DIFFERENTIAL EQUATION: A

differential equation is said to be of ordern, if the nth

derivative is the highest derivative in that equation.

Example: Order of d2y/dx2+dy/dx+y=2 is 2DEGREE OF A DIFFERENTIAL EQUATION: If the

given differential equation is a polynomial iny(n),

then the highest degree ofy(n) is defined as the degree

of the differential equation.

Example: Degree of (dy/dx)4+y=0 is 4

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SOLUTION OF A DIFFERENTIAL

EQUATION

SOLUTION: Any relation connecting the variables of anequation and not involving their derivatives, which satisfiesthe given differential equation is called a solution.

GENERAL SOLUTION: A solution of a differential equation

in which the number of arbitrary constant is equal to the orderof the equation is called a general or complete solution orcomplete primitive of the equation.

Example:y =Ax +B

PARTICULAR SOLUTION: The solution obtained by givingparticular values to the arbitrary constants of the generalsolution, is called a particular solution of the equation.

Example:y = 3x + 5

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Lecture-3

EXACT DIFFERENTIAL EQUATION

Let M(x,y)dx + N(x,y)dy = 0 be a first order

and first degree differential equation where M

and N are real valued functions for some x, y.

Then the equation Mdx + Ndy = 0 is said to be

an exact differential equation ifM/y=N/x

Example:

(2y sinx+cosy)dx=(x siny+2cosx+tany)dy

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Lecture-4

Working rule to solve an exact equation

STEP 1: Check the condition for exactness,

if exact proceed to step 2.

STEP 2: After checking that the equation isexact, solution can be obtained as

M dx+(terms not containing x) dy=c

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Lecture-5

INTEGRATING FACTOR

Let Mdx + Ndy = 0 be not an exact differentialequation. Then Mdx + Ndy = 0 can be madeexact by multiplying it with a suitable function

u is called an integrating factor.Example 1:ydx-xdy=0 is not an exact equation.

Here 1/x2 is an integrating factor

Example 2:y(x

2

y2

+2)dx+x(2-2x2

y2

)dy=0 is notan exact equation. Here 1/(3x3y3) is anintegrating factor

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Lecture-6

METHODS TO FIND INTEGRATING

FACTORS

METHOD 1: With some experience

integrating factors can be found by inspection.

That is, we have to use some known

differential formulae.

Example 1:d(xy)=xdy+ydx

Example 2:d(x/y)=(ydx-xdy)/y2

Example 3:d[log(x2+y2)]=2(xdx+ydy)/(x2+y2)

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Lecture-7

METHODS TO FIND INTEGRATING

FACTORS

METHOD 2: If Mdx + Ndy = 0 is a non-exact

but homogeneous differential equation and

Mx + Ny 0 then 1/(Mx + Ny) is anintegrating factor of Mdx + Ndy = 0.

Example 1:x2ydx-(x3+y3)dy=0 is a non-exact

homogeneous equation. Here I.F.=-1/y4

Example 2:y2dx+(x2-xy-y2)dy=0 is a non-exact

homogeneous equation. Here I.F.=1/(x2y-y3)

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Lecture-8

METHODS TO FIND INTEGRATING

FACTORSMETHOD 3: If the equation Mdx + Ndy = 0 is of the

form y.f(xy) dx + x.g(xy) dy = 0 and MxNy 0then 1/(MxNy) is an integrating factor of Mdx +

Ndy = 0.Example 1:y(x2y2+2)dx+x(2-2x2y2)dy=0 is non-exact

and in the above form. Here I.F=1/(3x3y3)

Example 2:(xysinxy+cosxy)ydx+(xysinxy-

cosxy)xdy=0 is non-exact and in the above form.

Here I.F=1/(2xycosxy)

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Lecture-9

METHODS TO FIND INTEGRATING FACTORS

METHOD 4: If there exists a continuous single

variable function f(x) such that M/y-N/x=Nf(x) then ef(x)dx is an integrating factor of

Mdx + Ndy = 0Example 1:2xydy-(x2+y2+1)dx=0 is non-exact and

M/y - N/x=N(-2/x). Here I.F=1/x2

Example 2:(3xy-2ay

2

)dx+(x

2

-2axy)=0 is non-exactand M/y - N/x=N(1/x). Here I.F=x

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Lecture-10

METHODS TO FIND INTEGRATING

FACTORS

METHOD 5: If there exists a continuous single

variable function f(y) such that

N/x - M/y=Mg(y) then eg(y)dy is an integratingfactor ofMdx + Ndy = 0

Example 1:(xy3+y)dx+2(x2y2+x+y4)dy=0 is a non-

exact equation and N/x - M/y=M(1/y). Here

I.F=yExample 2:(y4+2y)dx+(xy3+2y4-4x)dy=0 is a non-

exact equation and N/x - M/y=M(-3/y).HereI.F=1/y3

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Lecture-11

LEIBNITZ LINEAR EQUATION

An equation of the form y + Py = Q is called alinear differential equation.

Integrating Factor(I.F.)=epdx

Solution is y(I.F) = Q(I.F)dx+CExample 1:xdy/dx+y=logx. Here I.F=x and solution

is xy=x(logx-1)+C

Example 2:dy/dx+2xy=e

-x

.Here I.F=ex

and solutionis yex=x+C

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Lecture-12

BERNOULLIS LINEAR EQUATION

An equation of the form y + Py = Qyn is called aBernoullis linear differential equation. Thisdifferential equation can be solved by reducing it tothe Leibnitz linear differential equation. For this

dividing above equation by yn

Example 1: xdy/dx+y=x2y6.Here I.F=1/x5 andsolution is 1/(xy)5=5x3/2+Cx5

Example 2: dy/dx+y/x=y2xsinx. Here I.F=1/x and

solution is 1/xy=cosx+C

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Lecture-13

ORTHOGONAL TRAJECTORIES

If two families of curves are such that each memberof family cuts each member of the other family atright angles, then the members of one family areknown as the orthogonal trajectories of the otherfamily.

Example 1: The orthogonal trajectory of the family ofparabolas through origin and foci on y-axis isx2/2c+y2/c=1

Example 2: The orthogonal trajectory of rectangularhyperbolas is xy=c2

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PROCEDURE TO FIND ORTHOGONAL

TRAJECTORIES

Supposef (x ,y ,c) = 0 is the given family of

curves, where c is the constant.

STEP 1: Form the differential equation byeliminating the arbitrary constant.

STEP 2: Replace y by -1/y in the above

equation.STEP 3: Solve the above differential equation.

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Lecture-14

NEWTONS LAW OF COOLING

The rate at which the temperature of a hotbody decreases is proportional to thedifference between the temperature of the body

and the temperature of the surrounding air. (0)

Example: If a body is originally at 80oC and

cools down to 60o

C in 20 min.If thetemperature of the air is at 40oC then thetemperture of the body after 40 min is 50oC

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Lecture-15

LAW OF NATURAL GROWTH

When a natural substance increases in Magnitude as a

result of some action which affects all parts equally,

the rate of increase depends on the amount of the

substance present.N = k N

Example: If the number N of bacteria in a culture

grew at a rate proportional to N. The value of N was

initially 100 and increased to 332 in 1 hour. Then the

value of N after one and half hour is 605

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LAW OF NATURAL DECAY

The rate of decrease or decay of any substance

is proportion to N the number present at time.

N = -k N

Example:A radioactive substance disintegrates

at a rate proportional to its mass. When mass is

10gms, the rate of disintegration is 0.051gms

per day. The mass is reduced to 10 to 5gms in

136 days.